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Gorenstein model structures and generalized derived categories

Part of: Homological algebra Applied homological algebra and category theory Abelian categories

Published online by Cambridge University Press:  05 August 2010

James Gillespie  and
Mark Hovey
James Gillespie
Affiliation:
Department of Mathematics, Penn State Greater Allegheny, 4000 University Drive, McKeesport, PA 15132-7698, USA (jrg21@psu.edu)
Mark Hovey
Affiliation:
Department of Mathematics, Wesleyan University, Middletown, CT 06459, USA (hovey@member.ams.org)
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Abstract

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In a paper from 2002, Hovey introduced the Gorenstein projective and Gorenstein injective model structures on R-Mod, the category of R-modules, where R is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce: the Gorenstein flat model structure. The homotopy category with respect to each of these is called the stable module category of R. If such a ring R has finite global dimension, the graded ring R[x]/(x2) is Gorenstein and the three associated Gorenstein model structures on R[x]/(x2)-Mod, the category of graded R[x]/(x2)-modules, are nothing more than the usual projective, injective and flat model structures on Ch(R), the category of chain complexes of R-modules. Although these correspondences only recover these model structures on Ch(R) when R has finite global dimension, we can set R = ℤ and use general techniques from model category theory to lift the projective model structure from Ch(ℤ) to Ch(R) for an arbitrary ring R. This shows that homological algebra is a special case of Gorenstein homological algebra. Moreover, this method of constructing and lifting model structures carries through when ℤ[x]/(x2) is replaced by many other graded Gorenstein rings (or Hopf algebras, which lead to monoidal model structures). This gives us a natural way to generalize both chain complexes over a ring R and the derived category of R and we give some examples of such generalizations.

Keywords

Gorenstein ringderived categorymodel categorygeneralized chain complexstable module category

MSC classification

Secondary: 18E30: Derived categories, triangulated categories 18G35: Chain complexes 55U15: Chain complexes 55U35: Abstract and axiomatic homotopy theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 53 , Issue 3 , October 2010 , pp. 675 - 696
Copyright
Copyright © Edinburgh Mathematical Society 2010

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